The graph shows the distance a cyclist traveled in yards (y) as a function of time in seconds (x). The graph is divided into four segments; labeled P, Q, R, and S, respectively.


Which segment did the cyclist complete before stopping for a drink of water?

P
Q
R
S

Answer:

[tex]CD=3\sqrt{3}\ cm,\\ \\A_{ABC}=18\sqrt{3}\ cm^2[/tex]

Step-by-step explanation:

Triangle ABC is right triangle. Since  m∠ACB = 90°  and m∠ACD = 60°, you get m∠BCD = 90°-60°=30°.

Triangle BCD is rigth triangle, then the leg that is opposite to the angle of 30° is half of the hypotenuse. Thus,

[tex]BD=\dfrac{1}{2}BC=\dfrac{1}{2}\cdot 6=6\ cm.[/tex]

By the Pythagorean theorem,

[tex]CD^2=BC^2-BD^2=6^2-3^2=36-9=27,\\ \\CD=3\sqrt{3}\ cm.[/tex]

Then

[tex]CD^2=AD\cdot BD,\\ \\27=3\cdot AD,\\ \\AD=9\cm.[/tex]

Hypotenuse AB of the triangle ABC is equal to

[tex]AB=AD+BD=9+3=12\ cm.[/tex]

The area of the triangle ABC is

[tex]A_{ABC}=\dfrac{1}{2}\cdot AB\cdot CD=\dfrac{1}{2}\cdot 12\cdot 3\sqrt{3}=18\sqrt{3}\ cm^2.[/tex]

CD is √3 cm, and the area of right-angled triangle ABC, with ∠ACB = 90° and ∠ACD = 60°, is 9 cm², obtained by applying the Pythagorean Theorem and area formula.

The given triangle is right-angled at C (as per the given condition m∠ACB = 90°), and ∠ACD is 60°.

Thus, we know that ΔACD is a 30-60-90 triangle because the remaining angle of the triangle (∠ADC) would have to be 30° (since all angles in any triangle sum up to 180°).

In a 30-60-90 triangle, the ratio of the sides opposite to these angles respectively is 1 : √3 : 2. This tells us AC is 2 times the length of CD and AC is √3 times the length of AD.

We don't have the lengths of AC, AD, or CD yet, but we can solve for CD using the Pythagorean Theorem in triangle ABC. Since it's a right triangle, the square of the hypotenuse (AC in our case) is equal to the sum of the squares of the other two sides. We can express this as:

AC² = BC² - CD²

Substituting AC = 2CD into this equation, we get:

(2CD)² = 6² - CD²

Solving this equation yields CD = BC / (2 √3) = 6 / (2 √3) = √3 cm

So the length of CD is √3 cm.

Now let's find the area of triangle ΔABC. The area is typically given by the formula 0.5 * base * height.

In this case, BC is the base of triangle ABC, and CD is the height due to its perpendicular nature to side AB by the given condition (CD ⊥ AB). So, the calculation would go as follows:

Area = 0.5 * BC * CD = 0.5 * 6 cm * √3 cm = 3 √3 cm².

But by rationalizing the denominator, which is a common practice especially when having square roots in the denominator, we get Area = 3√3 * √3 * √3 cm² = 3 * 3 cm² = 9 cm².

Therefore, CD = √3 cm and the area of ΔABC is 9 cm².

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It's a quadratic function.

[tex]f(x)=ax^2+bx+c[/tex]

----------------------------------

[tex]f(x)=3x^2-8[/tex]

a = 3, b = 0, c = -8

a = 3 > 0, therefore the parabola open up.

The range of this function is [k, ∞).

k - second coordinate of the vertex (h, k)

[tex]h=\dfrac{-b}{2a},\ k=f(h)[/tex]

Substitute:

[tex]h=\dfrac{-0}{2(3)}=0\\\\k=f(0)=3(0^2)-8=0-8=-8[/tex]

Answer: The range is [tex][8,\ \infty)[/tex]

To find the range of the function ƒ(x) = 3x^2 - 8, we need to determine the set of all possible output values. The range of the function is all real numbers greater than or equal to the y-coordinate of the vertex, which is -8. Therefore, the range of the function ƒ(x) = 3x^2 - 8 is y ≤ -8.

To find the range of the function ƒ(x) = 3x2 − 8, we need to determine the set of all possible output values. Since the function is a quadratic function, its graph is a parabola. The range can be found by considering the vertex of the parabola and the direction it opens.

The vertex of the parabola is given by the formula x = -b/(2a), where a and b are the coefficients of the quadratic function. In this case, a = 3 and b = 0. So, the x-coordinate of the vertex is x = -0/(2*3) = 0.

Since the parabola opens upward (the coefficient of x2 is positive), the vertex represents the minimum value of the function. Therefore, the range of the function is all real numbers greater than or equal to the y-coordinate of the vertex. Plugging the x-coordinate (0) into the function gives us ƒ(0) = 3*(0)2 − 8 = -8. So, the range of the function ƒ(x) = 3x2 − 8 is y ≤ -8.

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Answer:

Step-by-step explanation:

When the second train leaves, the remaining distance between the trains is ...

... 600 km - (3 h)×(60 km/h) = 420 km

That distance will be covered at the speed of (60 +v) so the time it takes for the second train to cover that distance is ...

... 420/(60 +v)

The variable t represents the time since the first train left, so is 3 hours more than this time value. Hence ...

... t = 3 +420/(60 +v)

Solving for v, we have ...

... t -3 = 420/(60 +v) . . . . subtract 3

... 60 +v = 420/(t -3) . . . . multiply by (60+v)/(t -3)

... v = 420/(t -3) -60 . . . . subtract 60

For t = 7 ...

... v = 420/(7 -3) -60 = 105 -60 = 45

For t = 6 ...

... v = 420/(6 -3) -60 = 140 -60 = 80

Answer:

Correct choice is D

Step-by-step explanation:

Consider the exponential function [tex]R=10.5\cdot (0.535)^t.[/tex]

The exponential function of exponential decay can be written as

[tex]y=a\cdot (1-r)^x,[/tex]

where r is the decay rate.

Note that 1-0.535=0.465. The decimal 0.465 is 46.5% and this percent represents the rate of exponential decay.

Answer:

dont try B its wrong on edge :(

Step-by-step explanation:

i just took the text

Answer:

z-score for 900 points = -4

Step-by-step explanation:

To find the z scores

z=(x-mean)/standard deviation

so

z=(900-1500)/150

z= -600/150

Answer:

This may help you I think so

It is a nonlinear function.

When the points are graphed, they do not all lie on the same line. The lines connecting the points have positive slope, so the function is neither constant nor decreasing.

Answer: It is a nonlinear function.

Step-by-step explanation:

A function is said to be linear if the rate of change of dependent varaible (y) with respect to independent variable (x) is constant.

Rate of change in function [tex]=\dfrac{\text{Change in y}}{\text{Change in x}}[/tex]

The table gives the speeds recorded by the speedometer on Sully's car with respect to time. :

x                                        y

(seconds)                       (mph)

2                                     60

8                                    120

16                                   170

For x= 2 and x= 8 , the rate of change in function will be

[tex]\dfrac{\text{Change in y}}{\text{Change in x}}\\\\=\dfrac{120-60}{8-2}=\dfrac{60}{6}=10[/tex]

For x= 8 and x= 16 , the rate of change in function will be

[tex]=\dfrac{170-120}{16-8}=\dfrac{50}{8}=6.25[/tex]

But 10 ≠ 6.25.

⇒ Rate of change is not constant.

It means the function non- linear.

Also , it is not a constant function because values of y is not constant w.r.t to x.

It is not a decreasing function because values of y increase with increase in x.

Answer:

I have different options

Step-by-step explanation:

Im not sure because I dont have (l)(w)(h)

Answer:

A

Step-by-step explanation:

Function cannot have at least two points on same vertical line.

Vertical test fails at x = 1. So we need to eliminate either point (1, 1) or (1, 3).

There is option (1, 1), so A.

To represent a function, we must have unique y-values for each x-value. Thus, the point (1,1) should be removed. The correct answer is A).

To determine which ordered pair should be removed from the graph to represent a function, we need to check for uniqueness in the x-values. In a function, each x-value can only be associated with one y-value.

Let's examine the given points:

(1,1)

(3,2)

(5,2)

(7,4)

(1,3)

The x-values are: 1, 3, 5, 7, and 1.

Notice that the x-value 1 is associated with both (1,1) and (1,3). In a function, an x-value should have a unique y-value. Therefore, the point (1,1) should be removed from the graph to represent a function.

So, the correct option is A) (1,1).

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Hello, there! :)

Answer:

s=4

*The answer must have a positive sign.*

Step-by-step explanation:

First, you divide by 6 from both sides of an equation.

[tex]\frac{6s}{6}=\frac{24}{6}[/tex]

Then, you divide by the numbers from left to right.

[tex]24/6=4[/tex]

Final answer is s=4

Hope this helps!

Have a nice day! :)

:D

-Charlie

Thank you so much! :)

:D

9514 1404 393

Answer:

  (b)  y = -2.5x² -5.5x +1.5

Step-by-step explanation:

The table entry (0, 1.5) tells you that the y-intercept is 1.5. That eliminates choices C and D.

Substituting x=4 into the first equation (choice A) shows you it is incorrect:

  y = -5.5(4²) -2.5(4) +1.5 = -88 -10 +1.5 = -96.5 . . . not -60.5

This only leaves choice B, a choice confirmed by a quadratic regression using a graphing calculator (or spreadsheet).

   y = -2.5x² -5.5x +1.5

Final answer:

The equation in standard form of the parabola that models the values in the table is y = -2.3x^2 + 2.6x + 1.5

Explanation:

The equation in standard form of a parabola that models the values in the table is y = ax^2 + bx + c. To find the values of a, b, and c, we can use the given table:

XF(x)-22.501.54-60.5

Plugging in the values from the table into the equation, we get a system of three equations:

4a - 2b + c = 2.5

c = 1.5

16a + 4b + c = -60.5

Solving this system of equations, we find that a = -2.3, b = 2.6, and c = 1.5.

Therefore, the equation in standard form of the parabola that models the values in the table is y = -2.3x^2 + 2.6x + 1.5.

Answer:

y +5 = (-2/3)(x -4)

Step-by-step explanation:

The point-slope form of an equation of a line with slope m through point (h, k) is ...

... y -k = m(x -h)

You have (h, k) = (4, -5) and a slope the same as that of the parallel line, m = -2/3. Then the equation is ...

... y -(-5) = (-2/3)(x -4)

... y +5 = (-2/3)(x -4)

Answer:

ITS C: Multiplying by 15 is the same as multiplying by 20%

Step-by-step explanation:

The solution is Option C.

The equation of multiplying by 1/5 is the same as multiplying by 20%

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the equation be represented as A

Now , the value of A is

Cara has driven one-fifth of the total distance, d, to her destination

So , the distance traveled by Cara S = ( 1/5 )d

And , the distance traveled by Cara S = 0.2d

On simplifying the equation , we get

The value of ( 1/5 )d = d/5

The value of 0.2d = ( 20/100 )d

The value of 0.2d = 20 % of d

Therefore , the value of ( 1/5 )d = value of 0.2d

Hence , the equations are same

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Answer:

The area of the entire yellow region is 48 cm^2.

Step-by-step explanation:

See the attached image :)

Answer:

B. x > -1

Step-by-step explanation:

Subtract the left side:

... 0 < -6(1 -x) -2(x-5)

... 0 < -6 +6x -2x +10 . . . . eliminate parentheses

... 0 < 4 +4x . . . . . . . . . . . collect terms

... 0 < 1 +x . . . . . . . . . . . . . divide by 4

... -1 < x . . . . . . . . . . . . . . . add -1. Matches selection B.

Answer:

W'' = (2, -3)

Step-by-step explanation:

Reflection over the x-axis negates the y-coordinate and leaves the x-coordinate alone. The W becomes ...

... W' = (-2, -3)

Reflection over the y-axis negates the x-coordinate and leaves the y-coordinate alone. The W' becomes ...

... W'' = (2, -3)

Answer:

The coordinates of point W'' are (2,-3).

Step-by-step explanation:

The coordinates of given point are W(-2,3).

If point W is reflected over the x-axis, then

[tex](x,y)\rightarrow (x,-y)[/tex]

[tex]W(-2,3)\rightarrow W'(-2,-3)[/tex]

If point W' is reflected over the y-axis, then

[tex](x,y)\rightarrow (-x,y)[/tex]

[tex]W'(-2,-3)\rightarrow W''(2,-3)[/tex]

Therefore the coordinates of point W'' are (2,-3).

Answer:

w = 25

Step-by-step explanation:

The function A(w) = -(w-25)² +625 is in vertex form. The vertex (maximum) is at (w, A) = (25, 625). A width of 25 meters will maximize the area.

The location of the number k is -1.4 and the sum of 1.4 and k is 0

How to determine the positions of the numbers

From the question, we have the following parameters that can be used in our computation:

The number k and 1.4 are additive inverses.

This means that

k + 1.4 = 0

Evaluate the like terms

So, we have

k = -1.4

So, the location of the number k is -1.4 and the sum of 1.4 and k is 0

Answer:

16.0

Step-by-step explanation:

The mnemonic SOH CAH TOA reminds you ...

... Tan = Opposite/Adjacent

... tan(32°) = 10/AC

Multiplying by AC and dividing by the tangent gives you ...

... 10/tan(32°) = AC = 16.0

Answer:  The required length of AC is 16.1 units.

Step-by-step explanation:  We are given to find the length of side AC of triangle ABC.

From the figure, we note that

the triangle ABC is a right-angled triangle, where

m∠C = 90°, m∠A = 32° and BC = 10 units.

For the acute angle A, side AC is the base and side BC is the perpendicular.

So, from trigonometric ratios, we have

[tex]\tan m\angle A=\dfrac{perpendicular}{base}\\\\\\\Rightarrow \tan32^\circ=\dfrac{BC}{AC}\\\\\\\Rightarrow \tan32^\circ=\dfrac{10}{AC}\\\\\\\Rightarrow 0.62=\dfrac{10}{AC}\\\\\\\Rightarrow AC=\dfrac{10}{0.62}\\\\\Rightarrow AC=16.13.[/tex]

Rounding to nearest tenth, we get

AC = 16.1 units.

Thus, the required length of AC is 16.1 units.

Answer:

The correct answer is option C, Multiplying by 1/5 is the same as multiplying by 20%.

Step-by-step explanation:

The equation 1/5 = 1 divided by 5 = 0.2

and 0.2 can be expressed as 20%

Thus both the factor 1/5 and 20 % are same

Answer:

c is the answer

Step-by-step explanation:

Final answer:

Two circles that have the same center but different radii are similar because they share the same shape but differ in size. The answer, therefore, is D. They are similar because they are of the same shape but different sizes.

Explanation:

The question involves two circles that share the same center but have different radii. According to geometric principles, similar figures have the same shape but not necessarily the same size, while congruent figures have both the same shape and the same size. Since the two circles share the same center (co-centric circles) and therefore the same shape (both are circular) yet differ in size, due to their different radii, the correct answer is that they are similar. Hence, the correct option is:

D. They are similar because they are of the same shape but different sizes.

This addresses the concept that similarity in geometry is about shape, rather than size. If the circles were both identical in size and shape, they would be congruent; however, because their sizes vary, they cannot be congruent.

Answer: y = 3

Step-by-step explanation:

The objective is to get 0 on the left side to equal the 0 on the right side. I'd like to do this by making the sum of x^2 and y^2 equal to 10. If (x^2+y^2) = 10, the -10 on the left side will cancel it out to equal 0.

1^2 = 1

3^3 = 9

1 + 9 = 10

Therefore, x is 1 and y is 3.

x^2 + y^2 - 10 = 0

1^2 + 3^2 - 10 - 0

1     +     9  - 10 = 0

10              -10 = 0

0 = 0

Answer:

C) -4

Step-by-step explanation:

This is easy:)

54 - 4 = 50

50 - 4 = 46

46 - 4 = 42

42 - 4 = 38

Hope this helped u!!!

Good luck:))

The common difference for this arithmetic sequence is -4.

An arithmetic sequence is defined in two ways. It is a "sequence where the differences between every two successive terms.

The given arithmetic sequence is;

54, 50, 46, 42, 38

The difference first term and second term is;

[tex]\rm a_2-a_1= 50-54=-4[/tex]

The difference second term and third term is;

[tex]\rm a_3-a_2= 46-50=-4[/tex]

The difference third term and fourth term is;

[tex]\rm a_4-a_3=42-46=-4[/tex]

The difference fourth term and last term is;

[tex]\rm a_5-a_4=38-42=-4[/tex]

Hence, the common difference for this arithmetic sequence is -4.

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We can let the variables A and B stand for the length in inches of fish A and fish B, respectively.

If we assume each person bought fish having a total length of 1 inch per gallon of aquarium, then we can write equations ...

... 3A +2B = 33 . . . . . total length of Marta's fish

... 3A +4B = 45 . . . . . total length of Hank's fish

Subtracting the first equation from the second, we get ...

... 2B = 12

... B = 6 . . . . . divide by 2

Using this value in the first equation, we have ...

... 3A + 2·6 = 33

... 3A = 21 . . . . . . . . subtract 12

... A = 7 . . . . . . . . . . divide by 3

Fish A is 7 inches long; fish B is 6 inches long.

To determine the lengths of fish A and fish B, two simultaneous equations were formed using the information provided. Solving these equations resulted in finding that fish A is 7 inches long and fish B is 6 inches long.

The problem revolves around figuring out the length of fish A and fish B given the rule that each inch of fish requires 1 gallon of water.

Marta's aquarium holds 33 gallons and she bought 3 of fish A and 2 of fish B. Hank's aquarium holds 45 gallons and he bought 3 of fish A and 4 of fish B. Let's denote the length of fish A as 'a' inches and fish B as 'b' inches.

Marta's equation: 3a + 2b = 33 (1)

Hank's equation: 3a + 4b = 45 (2)

To find the length of fish A and B, we solve these simultaneous equations:

Fish A is 7 inches long, and fish B is 6 inches long.

(m, a, b, c) ∈ {(1, 196, 441, 210), (7, 28, 63, 30)}

m is a common factor of 196, 210, 441

The prime factors of those numbers are

... 196 = 2²×7²

... 210 = 2×3×5×7

... 441 = 3²×7²

7 is the only prime factor common to all the numbers. Hence the possible values of m are 1 and 7.

For m = 1, (a, b, c) = (196, 441, 210).

For m = 7, (a, b, c) = (196, 441, 210)/7 = (28, 63, 30).

Answer:

ave rate of change = k(5) - k(-3)

                                  ----------------

                                   5+3

Step-by-step explanation:

To find the average rate of change

ave rate of change = f(x2) - f(x1)

                                  ----------------

                                   x2-x1

We know that x2 = 5 and x1 = -3

ave rate of change = k(5) - k(-3)

                                  ----------------

                                   5--3

ave rate of change = k(5) - k(-3)

                                  ----------------

                                   5+3

The correct equation for the multiplication of means and extremes in the proportion 13/78 = 9/54 is C, which states 13 * 54 equals 78 * 9.

The equation that correctly shows the multiplication of the means and extremes in the proportion 13/78 = 9/54 is option C. This can be demonstrated by cross-multiplying the terms of the two ratios, where the product of the means (the two middle terms) is equal to the product of the extremes (the two outer terms). Therefore, the correct equation is 13 * 54 = 78 * 9.

To check, we can multiply the numbers:

Both products are equal, confirming that option C is the correct answer.

To find speed, divide total distance by total time:

Speed = 19.2 / 3 = 6.4 kilometers per hour

Answer:

The maximal number of smothies that Carol can make with the yogurt is 4.

Step-by-step explanation:

Divide the number of cups Carol has by the number of cups needed to make one smoothie.

[tex]1\dfrac{5}{8}:\dfrac{1}{3}=\dfrac{13}{8}\cdot \dfrac{3}{1}=\dfrac{39}{8}=4\dfrac{7}{8}.[/tex]

The maximal number of smothies that Carol can make with the yogurt is 4.

Answer:4

Step-by-step explanation:

1 5/8 multiply then add,you will get 13 Over 8.Keep,Change,recipical,And turn 1/3 To 3/1.Multiply,then divide because it will become an improper fraction.So divide then you will get 4 7/8.4 is a closer Number

Answer:

Step-by-step explanation:

The midsegment of a triangle is half the length of the side it is parallel to.

1. For a distance of 6.08 m between wall plates, the colar tie will be half that, or 3.04 m.

2. The perimeter of ΔXYZ is ...

... 10 + 12 + 14 = 36

The sides of ΔABC are half the length of the sides of ΔXYZ, so the perimeter of ΔABC is half the perimeter of ΔXYZ.

... perimeter of ΔABC = 36/2 = 18